M.A. Ashour
University of Alabama, Huntsville, USA
G.M. Norris
University of Nevada, Reno, USA
J.P. Singh
JP Singh and Associates, Richmond, California, USA
ABSTRACT: The paper demonstrates the effect of soil-structure interaction on the response of laterally loaded piles and drilled shafts in layered soil (sand and/or clay) and weak rock deposits. The paper also presents the capabilities of the Strain Wedge (SW) model technique and how it accounts for soil and pile property influence on the pile/shaft lateral response. The SW model has been validated and verified through several comparison studies with model- and full-scale lateral load tests. Several factors and features related to the problem of a laterally loaded isolated pile and pile group are covered by the SW model. For example, the nonlinear behavior of both soil and pile material, soil-pile interaction (i.e. the assessment of the p-y curves rather than the adoption of empirical ones), the potential of soil to liquefy, the interference among neighboring piles in a pile group, the pile cap contribution and the consideration of the pile/shaft type (short/intermediate and long) are considered in SW model analysis. The SW model analyzes the response of laterally loaded piles based on pile properties (pile stiffness, cross-sectional shape, pile-head conditions, etc.) as well as soil properties.
1 INTRODUCTION
The problem of a laterally loaded pile is often solved as a beam on an elastic foundation (BEF) involv-ing nonlinear modelinvolv-ing of the soil-pile interaction response (p-y curve). Currently employed p-y curve models were established/verified based on the results of field tests (Reese et al. 1974; Matlock 1970; and Reese & Welch 1975) and adjusted mathematically using empirical parameters to extrapolate beyond the soil’s specific field test conditions. While most design-ers prefer the p-y curve method as compared to elastic continuum or finite element analysis of later-ally loaded pile behavior, the profession has reached a state where it is time that closer scrutiny be given the traditional “Matlock-Reese” p-y curves used in the analysis. The traditional p-y curves were derived from a number of well-instrumented field tests that reflect a limited set of conditions. To consider these p-y curves as unique is questionable. The traditional p-y curve models are semi-empirical models in which soil response is characterized as independent nonlin-ear springs (Winkler springs) at discrete locations and do not account for a change in pile properties such as pile bending stiffness, pile cross-sectional shape, pile-head fixity and pile-pile-head embedment below the ground surface. Soil-pile interaction or p-y curve behavior is not unique but a function of both soil and pile proper-ties. Such influences can be considered using available
theoretical means (SW model formulation) that allows transformation of envisioned three-dimensional soil-pile interaction response to one-dimensional BEF parameters. As Terzaghi (1955) and Vesic (1961) stated, the subgrade modulus, Es(and, therefore, the p-y curve), is not just a soil but, rather, a soil-pile interaction (and, therefore, a pile property dependent) response (Figure 1). Kim et al. (2004) showed exper-imentally the significant effect of pile head fixity on the shape of the p-y curve in the same soil (Figure 2).
Figure 1. Subgrade reaction variation versus structure stiff-ness (Terzaghi 1955).
Figure 2. Effect of pile-head fixity on the p-y curve (Kim et al. 2004).
2 OVERVIEW OF THE STRAIN WEDGE MODEL BASIC CONCEPTS
The SW model parameters are related to an envisioned three-dimensional passive wedge of soil developing in front of the pile (Figure 3). The basic purpose of the SW model is to relate stress-strain-strength behavior of the layered soil in the wedge to one-dimensional BEF parameters. The SW model is, therefore, able to provide a theoretical link between the more complex three-dimensional soil-pile interaction and the sim-pler one-dimensional BEF characterization (Norris 1986). The previously noted correlation between the SW model response and BEF characterization reflects the following interdependence:
• the horizontal strain (ε) in the soil of the developing passive wedge in front of the pile to the deflection pattern (y versus depth, x) of the pile;
• the horizontal soil stress change (σh) in the devel-oping passive wedge to the soil-pile reaction (p) associated with BEF analysis; and
• the nonlinear variation in the Young’s modulus (E= (σh/ε) of the soil to the nonlinear variation in the modulus of subgrade reaction (Es= p/y) asso-ciated with BEF characterization as illustrated in detail by Norris (1986), Ashour et al. (1998) and Ashour & Norris (2000).
The reason for linking the SW model to BEF anal-ysis is to allow the appropriate selection of BEF parameters to solve the following differential equation:
EI indicates the pile bending stiffness, Q symbol-izes the axial load, y represents the lateral deflection of the pile and Es(x) is the modulus of subgrade reaction at depth x. It should be noted that axial load (Q−y) effect (i.e. induced moment) is part of the numerical analysis (Finite Difference Method) used to solve the
Figure 3. SW model force equilibrium, deflection and configuration.
BEF. Axial load along with bending moment are con-sidered in the calculation of normal stresses at any pile cross section that, in return, affect the pile cross section neutral axis and EI of that section.
The SW model is also a semi-empirical approach (AASHTO 2007) because, while based on theoretical concepts, stress-strain characterization is formulated from triaxial test behavior. The SW model yields suc-cessive points on the p-y curves caused by a change in the modulus of subgrade reaction, Es(x), profile
with increasing soil strain, ε (considered constant with depth in the upper passive wedge). Such an assumption is also valid with the short pile deflection pattern (Fig-ure 8) where the value of soil strain would be the same along the pile length for the upper and lower passive wedges. However, this is not the case with the deflected portion of the long and intermediate pile below the first zero-deflection point (lower passive wedge) shown in Figure 3a. At any increment of lateral loading, the lower passive soil wedge of the long and intermediate pile maintains different values of soil strain as shown in Figure 3a.
where is a parameter that relates the deflection angle (δ) to the soil strain ε in soil sublayer (i).
An effective stress (ES) analysis is also employed with clay as well as sand. The ES analysis for clay includes the development of excess porewater pressure with undrained loading based on Skempton’s equation (1954). By using an ES analysis with clay, the three-dimensional strain wedge geometry (Figure 3b) can be defined based on the more appropriate mobilized effective stress and friction angle, ϕm. The relation-ship between the normally consolidated clay undrained shear strength, Su, ϕ, and σhand the vertical effective stress, σvo, is presented in Ashour et al. (1998).
The SW model can handle the problem of multi-ple soil layers of different type. The soil profile and the loaded pile are divided into sublayers and seg-ments of constant thickness, as shown in Figure 3c.
Each sublayer of soil is considered to behave as a uni-form, isotropic, homogenous material and to have its own properties according to the sublayer location and soil type. The depth, h, of the passive wedge is con-trolled by the stability of the pile under the current pile head load. The effects of soil and pile properties are part of the soil-pile reaction along the pile length as reflected by the Young’s modulus of the soil (E), the stress level in the soil (SL), the pile deflection pat-tern (y vs. x or δ), and the BEF modulus of subgrade reaction (Es) between the pile segment and each soil sublayer (Figure 3c). To account for the interaction between soil layers and between the soil and pile, the deflected length of the pile is considered to be a con-tinuous beam of different short segments each with a uniform load resulting from the nonlinear Essupports from that sublayer (Figure 3c).
3 CHARACTERIZATION OF SOIL-PILE INTERACTION (p-y) OF A LATERALLY LOADED PILE
Corresponding to the horizontal equilibrium of a slice (a soil sublayer i) of the passive wedge at depth x (Figure 3b), the relationship between the soil-pile
reaction (p) and the horizontal and shear stress changes
σhand τ at horizontal strain ε at a certain depth can be expressed as
where the face width (BC) of the wedge at depth x is in relation to the pile width, D.
As presented by Ashour and Norris (2000), the ultimate values of p in sand and clay soils are deter-mined as
S1and S2, on the other hand, are shape factors that are 0.75 and 0.5, respectively, for a circular pile cross section, and 1.0 for a square pile (Briaud et al. 1984).
By combining the equations of the developing passive wedge geometry and the stress level with the above relationship, one finds that
Here the parameter A is a function of pile and wedge dimensions, applied stresses, and soil properties.
A separate expression for the stress level (SL) vs. ε gives the shape of the associated triaxial test normal-ized stress (SL) versus axial strain curve (Ashour et al.
1998). Suis the undrained shear strength of clay.
The level of mobilization of the passive wedge in a sublayer depends on SL of the soil in the wedge and the shear resistance, τ, along the pile sides. The val-ues of σvo, SL and τ vary according to drained sand or undrained clay conditions of that sublayer. The side shear stress in sand (Eqn 10) is a function of the mobilized side shear friction angle, ϕs, that reaches its ultimate value (ϕs= ϕ) prior to that of the mobi-lized friction angle, ϕm, of the sand in the wedge (i.e.
SLtand SL). The level of shear stress (SLt) along the pile sides (as in a direct or simple shear test) differs from the stress level (SL) of the soil in the wedge in front of the pile (as in the triaxial compression test with the horizontal direction in the field representing the axial direction in the lab).
Figure 4a. Effect of pile bending stiffness on the p-y curve at 0.915-m depth (soft clay, Sabine River test).
4 EFFECT OF SOIL AND PILE PROPERTIES ON THE p-y CURVE
The influence of pile properties (such as pile bend-ing stiffness, pile-head conditions, pile cross-sectional shape and pile-head embedment depth), and the effect of a change in the neighboring soil (above and/or below the given sublayer) on the nature of the result-ing p-y curve can be demonstrated via the SW model approach.
Based on SW model analysis, pile properties have a significant effect on the shape and geometry of the developing passive wedge and, hence, the values of pultand Aultin flow-around failure. In order to address this issue, consider two piles of the same diame-ter (D= 0.33 m, original EI = 31300 kN-m2) driven in soft clay (Matlock 1970) but of different bending stiffnesses (different materials). Figure 4a presents the free-head SW model p-y curves at 0.915 m below the ground surface for different EI values. It is noted that the ultimate resistance of soil-pile reaction is con-trolled by the soil-pile combination (Eqns 9 and 11, pult= 14.35 kPa). A very stiff pile (10 EI) in this soft clay does not interact well with the soil, and a deep and large passive wedge at higher stress levels (SL and SLt) quickly develops. Consequently, as Ai(given by Eqn 11) reaches its ultimate value, flow-around failure occurs at this depth and the soil-pile reaction, p, ceases at a value less than pult(Eqn 11) (Ashour and Norris 2000).
Reducing the bending stiffness of the pile to that of the original steel pipe pile (EI) yields an increase in Ai
(compared to the first case) and an increase in the range of soil-pile interaction until flow around failure again occurs at A= Ault for p < pult. A greater reduction in pile stiffness (similar to a R/C pile of 0.1EI) increases the ductility of the p-y curve resulting in approximately the same value of p at flow around failure (A= Ault).
However, for a very flexible pile (timber pile of 0.01EI) in this soft clay, very large deflection is required before the soil-pile reaction reaches pult at A= Ault. This is
Figure 4b. Effect of pile bending stiffness on the p-y curve at 1.83-m depth (Mustang island test site).
Table 1. Properties of soils used in the comparisons.
Unit Wt, φ *Su
Soil Type γ(kN/m3) E50% (degree) (kN/m2)
Loose Sand 16.5 0.005 30
Dense Sand 19.6 0.0025 40
Soft Clay 15.7 0.015 24
Stiff clay 19.6 0.005 86
because of the very slow growth of the passive wedge and parameter A.
Figure 4b presents the interaction between pile and sand at a depth of 1.83 m for conditions similar to the Mustang Island test (Cox et al. 1974). Changing the pile stiffness results in very different p-y curves.
Because the surrounding sand is dense, increasing the pile stiffness causes the p-y curve to become stiffer.
The p-y curve in the sand would cease to grow due to the development of a plastic hinge (yield moment) well before any flow-around failure. Note that the effect of yield moment is shown only for the Mustang Island test and SW p-y curves. Linear and nonlinear pile material models are employed in the SW model analysis. This includes elastic-plastic steel model and stress-strain model for confined concrete (Ashour et al 2001).
5 THE NON-UNIQUENESS OF THE p-y CURVE 5.1 Effect of pile bending stiffness on the p-y curve Two free-head piles (stiff and flexible), 0.305 m in diameter and 12 m long with different stiffnesses (31300 kN-m2and 4300 kN-m2), are used to further demonstrate the effect of pile stiffness on the nature of the p-y curve. Both piles are assumed to be driven in loose and dense sand and soft and stiff clay (Table 1).
The SW model is used to determine the values of the p-y curves at 1.22 m below the pile head.
Figure 5 shows the effect of pile stiffness on the SW model predicted p-y curves of a free-head pile in soft
Figure 5. Effect of pile stiffness on the p-y curve.
Figure 6. Effect of pile-head fixity on the p-y curve.
and stiff clay at 1.22 m depth. As expected, the stiff pile in stiff clay exhibits the highest soil resistance.
Pile stiffness has a greater effect on the p-y response in stiff clay. Changing the value of pile stiffness in soft clay has only a slight influence on the characteristic of the p-y curve as long as the soil has not failed.
5.2 Effect of pile-head conditions on the p-y curve Figure 6 shows the effect of pile-head conditions (free or fixed-head) as a significant factor in the SW model analysis that is affecting the shape of the developed p-y curve. Note that the fixed head p-p-y curve in stiff and soft clay (Figure 11) reaches pult at lower deflection (and pressure) than that of the free head p-y curve.
This is the result of the development of a larger passive wedge for the fixed head case at the same value of soil strain, ε (Ashour and Norris 2000). This is also in agreement with the results obtained by Kim et al. 2004 and shown in Figure 2.
5.3 Effect of pile cross-sectional shape on the p-y curve in sand
The SW model was used to assess the p-y curves at a 1.22-m depth in sand and clay of two reinforced con-crete piles which are assumed to have the same bending
Figure 7. Effect of pile-cross sectional shape on the p-y curve in sand.
Figure 8. Deflection patterns of long, intermediate and short piles.
stiffness of 11500 kN-m2. The first pile has a square cross-section of 0.305-m width, while the second pile has a circular cross-section of 0.305-m diameter. The only difference between the two piles is their cross-sectional shapes. As shown in Figure 7, the square pile in loose and dense sand exhibits a soil-pile resistance higher than that of the circular pile.
5.4 Effect of pile length on the p-y curve
Pile is defined as “a long pile” for L/T >= 4. L is the pile length and T is the soil-pile relative stiffness defined as (EI/f)0.2for sand and normally consolidated clay, where f is the coefficient of subgrade reaction (F/L3). The value of relative stiffness, T , varies with EI and f . For a short pile, the bending stiffness (EI) in the analysis could have a constant value (linear elastic) as a result of small flexural deformations.
The coefficient of subgrade reaction, f , varies with level of deflection and decreases with increasing lateral load. The chart attributable to Terzaghi (DM 7.2, NAV-FAC 1982) provides average (design) values of f as a function of the sand’s relative density. The pile behaves as an “intermediate” pile when [4 > (L/T) > 2]. If an intermediate or short pile is analyzed as a long pile
Figure 9. Effect of pile length on the shape of p-y curve at 3.3 m below ground.
Figure 10. Modeling large diameter drilled shaft with vertical as well as horizontal soil-shaft resistance.
(e.g. using p-y curves of long piles), an overestimated (stiffer) lateral pile response could be obtained.
Short, intermediate and long pile classifications are based on pile properties (i.e. length, diameter and bending stiffness) and soil stiffness. The traditional (Matlock-Reese) p-y curves for sand and clay were developed from full scale long pile test data (Mat-lock 1970, and Reese et al. 1974). The development of the mobilized three-dimensional passive soil wedges along the deflected length of short, intermediate and long piles was experimentally observed by Hughes et al. (1978).
Figure 11. Lateral response of large diameter shaft at the Taiwan test (Brown et al. 2001).
Four different values for the pile length (15, 18, 23 and 30 m) are used with a 1.5-m-diameter shaft embed-ded in the soil profile shown in Figure 9. The p-y curve at 3.3-m depth below the pile head changes accord-ing to the pile characterization (short, intermediate or long). Such behavior reflects the influence of soil-pile interaction and pile type on the associated shape of the p-y curve.
5.5 Effect of vertical side shear resistance (large diameter shafts)
Since the traditional p-y curve have been developed using lateral load tests performed on long slender piles, the vertical shear resistance (Vv) acting along the pile or shaft perimeter has no significant influence on the lateral response of shafts and piles with diameters less than 0.91 m. However, Vvcontributes significantly to the lateral capacity of large diameter shafts. The SW model accounts for the horizontal and vertical shear resistance (Vhand Vv) acting along the sides of the large diameter shafts in addition to base resistance (Ashour et al 2004a). Figure 11 shows the contribu-tion of the vertical side shear resistance to the lateral resistance of the 1.5-m diameter shaft tested at the Taiwan test (Brown et al. 2001).
6 PILE/SHAFT GROUP INCLUDING THE CAP EFFECT
As presented by Ashour et al. (2004b), the SW model allows the assessment of the mobilized group action among the piles in a group with no need for P-multipliers to be assumed. The interference (over-lapping) among neighboring passive soil wedges is determined along the piles at any kevel of loading.
The evaluation of the group action of pile group in the SW model analysis accounts for front and trans-verse pile spacing, soil types, level of lateral loading,
Figure 12. Simplified modeling of a pile group with a pile cap in the SW model.
and the depth of pile interference all of which are not considered in the p-multiplier technique. Several full-and model-scale case have been used to validate the laterally loaded pile group modeling in the SW model technique (WSDOT 2007, Ashour et al. 2004b, Rollins et al. 2005).
The SW model analyzes the pile cap as an element of the whole pile group foundation system (Figure 12) that is influenced by the pile head stiffness and the type of pile-head fixity. Front passive soil and side shear resistance for pile cap lateral movement is incor-porated in the analysis. Figure 13 shows the varying contribution of the pile cap to the lateral resistance of the deep foundation system that is caused by free-and fixed-head shaft-cap connection at the same lateral displacement.
The lateral response of deep foundation shown in Figure 13 is obtained from 2-ft diameter 3× 3 pile
The lateral response of deep foundation shown in Figure 13 is obtained from 2-ft diameter 3× 3 pile